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# Shape Functions (shape + function)

Distribution by Scientific Domains

Kinds of Shape Functions

 element shape function finite element shape function linear shape function

## Selected Abstracts

### Shape functions for polygonal domains with interior nodes

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2004
Elisabeth Anna Malsch
Abstract The presented formulation follows in a series of publications which outline a method for constructing test functions which satisfy essential edge conditions exactly. The method promises a complete solution, satisfying all of the requirements of a Ritz coordinate function. The influence of interior points on the domain solution is included in this construction. Similar to conformal bubble functions, the test functions are zero along the boundary and single valued only at the points they describe. Unlike the bubble function construction, the interior points can be located at any desired point in the domain. The resulting set of trial functions can satisfy the required global conditions including the exact reproduction of constant and linear fields. Copyright © 2004 John Wiley & Sons, Ltd. [source]

### An improved meshless collocation method for elastostatic and elastodynamic problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 8 2008
P. H. Wen
Abstract Meshless methods for solving differential equations have become a promising alternative to the finite element and boundary element methods. In this paper, an improved meshless collocation method is presented for use with either moving least square (MLS) or compactly supported radial basis functions (RBFs). A new technique referred to as an indirect derivative method is developed and compared with the direct derivative technique used for evaluation of second-order derivatives and higher-order derivatives of the MLS and RBF shape functions at the field point. As the derivatives are obtained from a local approximation (MLS or compact support RBFs), the new method is computationally economical and efficient. Neither the connectivity of mesh in the domain/boundary nor integrations with fundamental/particular solutions is required in this approach. The accuracy of the two techniques to determine the second-order derivative of shape function is assessed. The applications of meshless method to two-dimensional elastostatic and elastodynamic problems have been presented and comparisons have been made with benchmark analytical solutions. Copyright © 2007 John Wiley & Sons, Ltd. [source]

### Error estimation in a stochastic finite element method in electrokinetics

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2010
S. Clénet
Abstract Input data to a numerical model are not necessarily well known. Uncertainties may exist both in material properties and in the geometry of the device. They can be due, for instance, to ageing or imperfections in the manufacturing process. Input data can be modelled as random variables leading to a stochastic model. In electromagnetism, this leads to solution of a stochastic partial differential equation system. The solution can be approximated by a linear combination of basis functions rising from the tensorial product of the basis functions used to discretize the space (nodal shape function for example) and basis functions used to discretize the random dimension (a polynomial chaos expansion for example). Some methods (SSFEM, collocation) have been proposed in the literature to calculate such approximation. The issue is then how to compare the different approaches in an objective way. One solution is to use an appropriate a posteriori numerical error estimator. In this paper, we present an error estimator based on the constitutive relation error in electrokinetics, which allows the calculation of the distance between an average solution and the unknown exact solution. The method of calculation of the error is detailed in this paper from two solutions that satisfy the two equilibrium equations. In an example, we compare two different approximations (Legendre and Hermite polynomial chaos expansions) for the random dimension using the proposed error estimator. In addition, we show how to choose the appropriate order for the polynomial chaos expansion for the proposed error estimator. Copyright © 2009 John Wiley & Sons, Ltd. [source]

### A finite element formulation for thermoelastic damping analysis

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2009
Enrico Serra
Abstract We present a finite element formulation based on a weak form of the boundary value problem for fully coupled thermoelasticity. The thermoelastic damping is calculated from the irreversible flow of entropy due to the thermal fluxes that have originated from the volumetric strain variations. Within our weak formulation we define a dissipation function that can be integrated over an oscillation period to evaluate the thermoelastic damping. We show the physical meaning of this dissipation function in the framework of the well-known Biot's variational principle of thermoelasticity. The coupled finite element equations are derived by considering harmonic small variations of displacement and temperature with respect to the thermodynamic equilibrium state. In the finite element formulation two elements are considered: the first is a new 8-node thermoelastic element based on the Reissner,Mindlin plate theory, which can be used for modeling thin or moderately thick structures, while the second is a standard three-dimensional 20-node iso-parametric thermoelastic element, which is suitable to model massive structures. For the 8-node element the dissipation along the plate thickness has been taken into account by introducing a through-the-thickness dependence of the temperature shape function. With this assumption the unknowns and the computational effort are minimized. Comparisons with analytical results for thin beams are shown to illustrate the performances of those coupled-field elements. Copyright © 2008 John Wiley & Sons, Ltd. [source]

### Singularity extraction technique for integral equation methods with higher order basis functions on plane triangles and tetrahedra

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2003
Seppo Järvenpää
Abstract A numerical solution of integral equations typically requires calculation of integrals with singular kernels. The integration of singular terms can be considered either by purely numerical techniques, e.g. Duffy's method, polar co-ordinate transformation, or by singularity extraction. In the latter method the extracted singular integral is calculated in closed form and the remaining integral is calculated numerically. This method has been well established for linear and constant shape functions. In this paper we extend the method for polynomial shape functions of arbitrary order. We present recursive formulas by which we can extract any number of terms from the singular kernel defined by the fundamental solution of the Helmholtz equation, or its gradient, and integrate the extracted terms times a polynomial shape function in closed form over plane triangles or tetrahedra. The presented formulas generalize the singularity extraction technique for surface and volume integral equation methods with high-order basis functions. Numerical experiments show that the developed method leads to a more accurate and robust integration scheme, and in many cases also a faster method than, for example, Duffy's transformation. Copyright © 2003 John Wiley & Sons, Ltd. [source]

### Unequally spaced non-periodic B-spline finite strip method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2003
Chang-Koon Choi
Abstract The unequally spaced non-periodic B-spline finite strip method (FSM) is presented. The motivation to investigate the irregularly spaced interior nodes in the longitudinal direction of strip is to generalize the concept of non-periodic B-spline FSM and to improve the general accuracy of the stress evaluation in the region of high stress gradients. In the present paper, the unequally spaced non-periodic B3-spline series with multiple knots at the boundary are introduced for the interpolation of displacement and description of geometry in the formulation of isoparametric spline FSM. The use of multiple knots at the boundary makes the shape function satisfy the Kronecker delta properties at the boundary. The unequally spaced B-spline FSM is applied to the stress-reduced shell problem with six degrees of freedom per node. The main purpose of this study is to find a way of ensuring that the geometry of strip is appropriately approximated when the interior nodes of the strip are not regularly spaced along the longitudinal direction. Some numerical results have been compared with those of the previous studies to evaluate the accuracy and efficiency of this method. Copyright © 2003 John Wiley & Sons, Ltd. [source]

### A reproducing kernel method with nodal interpolation property

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 7 2003
Jiun-Shyan Chen
Abstract A general formulation for developing reproducing kernel (RK) interpolation is presented. This is based on the coupling of a primitive function and an enrichment function. The primitive function introduces discrete Kronecker delta properties, while the enrichment function constitutes reproducing conditions. A necessary condition for obtaining a RK interpolation function is an orthogonality condition between the vector of enrichment functions and the vector of shifted monomial functions at the discrete points. A normalized kernel function with relative small support is employed as the primitive function. This approach does not employ a finite element shape function and therefore the interpolation function can be arbitrarily smooth. To maintain the convergence properties of the original RK approximation, a mixed interpolation is introduced. A rigorous error analysis is provided for the proposed method. Optimal order error estimates are shown for the meshfree interpolation in any Sobolev norms. Optimal order convergence is maintained when the proposed method is employed to solve one-dimensional boundary value problems. Numerical experiments are done demonstrating the theoretical error estimates. The performance of the method is illustrated in several sample problems. Copyright © 2003 John Wiley & Sons, Ltd. [source]

### Moving kriging interpolation and element-free Galerkin method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2003
Lei Gu
Abstract A new formulation of the element-free Galerkin (EFG) method is presented in this paper. EFG has been extensively popularized in the literature in recent years due to its flexibility and high convergence rate in solving boundary value problems. However, accurate imposition of essential boundary conditions in the EFG method often presents difficulties because the Kronecker delta property, which is satisfied by finite element shape functions, does not necessarily hold for the EFG shape function. The proposed new formulation of EFG eliminates this shortcoming through the moving kriging (MK) interpolation. Two major properties of the MK interpolation: the Kronecker delta property (,I(sJ)=,IJ) and the consistency property (,In,I(x)=1 and ,In,I(x)xIi=xi) are proved. Some preliminary numerical results are given. Copyright © 2002 John Wiley & Sons, Ltd. [source]

### The natural volume method (NVM): Presentation and application to shallow water inviscid flows

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2009
R. Ata
Abstract In this paper a fully Lagrangian formulation is used to simulate 2D shallow water inviscid flows. The natural element method (NEM), which has been used successfully with several solid and fluid mechanics applications, is used to approximate the fluxes over Voronoi cells. This particle-based method has shown huge potential in terms of handling problems involving large deformations. Its main advantage lies in the interpolant character of its shape function and consequently the ease it allows with respect to the imposition of Dirichlet boundary conditions. In this paper, we use the NEM collocationally, and in a Lagrangian kinematic description, in order to simulate shallow water flows that are boundary moving problems. This formulation is ultimately shown to constitute a finite-volume methodology requiring a flux computation on Voronoi cells rather than the standard elements, in a triangular or quadrilateral mesh. St Venant equations are used as the mathematical model. These equations have discontinuous solutions that physically represent the existence of shock waves, meaning that stabilization issues have thus been considered. An artificial viscosity deduced from an analogy with Riemann solvers is introduced to upwind the scheme and therefore stabilize the method. Some inviscid bidimensional flows were used as preliminary benchmark tests, which produced decent results, leading to well-founded hopes for the future of this method in real applications. Copyright © 2008 John Wiley & Sons, Ltd. [source]

### Microwave imaging of parallel perfectly conducting cylinders

INTERNATIONAL JOURNAL OF IMAGING SYSTEMS AND TECHNOLOGY, Issue 6 2000
Anyong Qing
This paper considers microwave imaging of parallel perfectly conducting cylinders using a solution of the scattering problem by the point-matching method. A cubic B-spline, real-coded genetic algorithm and an adaptive hybrid algorithm are proposed to solve the inverse problem. Previous shape functions in trigonometric series with arbitrary coefficients are nondefinite, which intensify the ill-posedness and slow the early time convergence of the algorithm. A novel shape function based on cubic B-splines is developed and the real-coded genetic algorithm is modified accordingly. Numerical simulation examples show that the early time convergence of the real-coded genetic algorithm is improved significantly. Next, the adaptive hybrid algorithm is developed to improve the late time convergence of the cubic B-spline real-coded genetic algorithm. © 2001 John Wiley & Sons, Inc. Int J Imaging Syst Technol 11, 365,371, 2000 [source]

### A neural-network-based model for 2D microwave imaging of cylinders

INTERNATIONAL JOURNAL OF RF AND MICROWAVE COMPUTER-AIDED ENGINEERING, Issue 5 2004
Kun-Chou Lee
Abstract In this article, a neural network with radial-basis functions (RBF-NN) is applied to microwave imaging of cylinders. Initially, the shape function of the target cylinder is expanded by a Fourier series. The RBF-NN is trained by some direct-scattering data sets and thus can predict the images of the target cylinders. © 2004 Wiley Periodicals, Inc. Int J RF and Microwave CAE 14, 398,403, 2004. [source]

### Nonparametric maximum likelihood estimation for shifted curves

JOURNAL OF THE ROYAL STATISTICAL SOCIETY: SERIES B (STATISTICAL METHODOLOGY), Issue 2 2001
Birgitte B. Rønn
The analysis of a sample of curves can be done by self-modelling regression methods. Within this framework we follow the ideas of nonparametric maximum likelihood estimation known from event history analysis and the counting process set-up. We derive an infinite dimensional score equation and from there we suggest an algorithm to estimate the shape function for a simple shape invariant model. The nonparametric maximum likelihood estimator that we find turns out to be a Nadaraya,Watson-like estimator, but unlike in the usual kernel smoothing situation we do not need to select a bandwidth or even a kernel function, since the score equation automatically selects the shape and the smoothing parameter for the estimation. We apply the method to a sample of electrophoretic spectra to illustrate how it works. [source]

### Amplitude,shape approximation as an extension of separation of variables

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2008
N. Parumasur
Abstract Separation of variables is a well-known technique for solving differential equations. However, it is seldom used in practical applications since it is impossible to carry out a separation of variables in most cases. In this paper, we propose the amplitude,shape approximation (ASA) which may be considered as an extension of the separation of variables method for ordinary differential equations. The main idea of the ASA is to write the solution as a product of an amplitude function and a shape function, both depending on time, and may be viewed as an incomplete separation of variables. In fact, it will be seen that such a separation exists naturally when the method of lines is used to solve certain classes of coupled partial differential equations. We derive new conditions which may be used to solve the shape equations directly and present a numerical algorithm for solving the resulting system of ordinary differential equations for the amplitude functions. Alternatively, we propose a numerical method, similar to the well-established exponential time differencing method, for solving the shape equations. We consider stability conditions for the specific case corresponding to the explicit Euler method. We also consider a generalization of the method for solving systems of coupled partial differential equations. Finally, we consider the simple reaction diffusion equation and a numerical example from chemical kinetics to demonstrate the effectiveness of the method. The ASA results in far superior numerical results when the relative errors are compared to the separation of variables method. Furthermore, the method leads to a reduction in CPU time as compared to using the Rosenbrock semi-implicit method for solving a stiff system of ordinary differential equations resulting from a method of lines solution of a coupled pair of partial differential equations. The present amplitude,shape method is a simplified version of previous ones due to the use of a linear approximation to the time dependence of the shape function. Copyright © 2007 John Wiley & Sons, Ltd. [source]

### A marker method for the solution of the damped Burgers' equation

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2006
Jerome L. V. Lewandowski
Abstract A new method for the solution of the damped Burgers' equation is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some details. The marker method is applicable to a general class of nonlinear dispersive partial differential equations. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 [source]

### Semilinear parabolic problem with nonstandard boundary conditions: Error estimates

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2003
Marián Slodi
Abstract We study a semilinear parabolic partial differential equation of second order in a bounded domain , , ,N, with nonstandard boundary conditions (BCs) on a part ,non of the boundary ,,. Here, neither the solution nor the flux are prescribed pointwise. Instead, the total flux through ,non is given, and the solution along ,non has to follow a prescribed shape function, apart from an additive (unknown) space-constant ,(t). We prove the well-posedness of the problem, provide a numerical method for the recovery of the unknown boundary data, and establish the error estimates. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 167,191, 2003 [source]

### Molecular shapes from small-angle X-ray scattering: extension of the theory to higher scattering angles

ACTA CRYSTALLOGRAPHICA SECTION A, Issue 2 2009
V. L. Shneerson
A low-resolution shape of a molecule in solution may be deduced from measured small-angle X-ray scattering I(q) data by exploiting a Hankel transform relation between the coefficients of a multipole expansion of the scattered amplitude and corresponding coefficients of the electron density. In the past, the radial part of the Hankel transform has been evaluated with the aid of a truncated series expansion of a spherical Bessel function. It is shown that series truncation may be avoided by analytically performing the radial integral over an entire Bessel function. The angular part of the integral involving a spherical harmonic kernel is performed by quadrature. Such a calculation also allows a convenient incorporation of a molecular hydration shell of constant density intermediate between that of the protein and the solvent. Within this framework, we determine the multipole coefficients of the shape function by optimization of the agreement with experimental data by simulated annealing. [source]

### Seismic interaction in electrical substation equipment connected by non-linear rigid bus conductors

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, Issue 2 2007
Junho Song
Abstract An important element within the power transmission lifeline is the electrical substation that typically consists of a complex set of equipment items interconnected through conductor buses or cables. If the connections are not sufficiently flexible, significant dynamic interaction may occur between the connected equipment items during a seismic event, which may result in damage and loss of the equipment. This paper investigates the interaction effect between electrical substation equipment items connected by non-linear rigid bus conductors. The equipment items are modelled as single-degree-of-freedom oscillators by use of appropriate shape functions. The hysteretic behaviours of rigid bus connectors are described by differential equation models fitted to experimental data. An efficient non-linear random vibration method is used to quantify the seismic interaction effect of the connected equipment items. Based on the developed analytical models and method, the effect of interaction in the connected equipment system is investigated through extensive parametric studies. The results lead to practical charts and guidelines for the seismic design of interconnected electrical substation equipment. Copyright © 2006 John Wiley & Sons, Ltd. [source]

### Development of the extended parametric meshless Galerkin method to predict the crack propagation path in two-dimensional damaged structures

FATIGUE & FRACTURE OF ENGINEERING MATERIALS AND STRUCTURES, Issue 7 2009
M. MUSIVAND-ARZANFUDI
ABSTRACT The parametric meshless Galerkin method (PMGM) enhances the promising features of the meshless methods by utilizing the parametric spaces and parametric mapping, and improves their efficiency from the practical viewpoints. The computation of meshless shape functions has been usually a time-consuming and complicated task in the meshless methods. In the PMGM, the meshless shape functions are mapped from the parametric space to the physical space, and therefore, the necessary computational time to generate the meshless shape functions is saved. The extended parametric meshless Galerkin method (X-PMGM) even improves the parametric property of the PMGM by incorporating the partition of unity concepts. In this paper, the development of the X-PMGM is extended by incorporating a crack-tip formulation in X-PMGM for fracture analysis and prediction of crack propagation path in the damaged structures. In this formulation, meshless shape functions are enriched by discontinuous enrichment function as well as crack-tip enrichment functions. The obtained results show that the predicted crack growth path is in good agreement with the experimental results. [source]

### Certified solutions for hydraulic structures using the node-based smoothed point interpolation method (NS-PIM)

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 15 2010
J. Cheng
Abstract A meshfree node-based smoothed point interpolation method (NS-PIM), which has been recently developed for solid mechanics problems, is applied to obtain certified solutions with bounds for hydraulic structure designs. In this approach, shape functions for displacements are constructed using the point interpolation method (PIM), and the shape functions possess the Kronecker delta property and permit the straightforward enforcement of essential boundary conditions. The generalized smoothed Galerkin weak form is then applied to construct discretized system equations using the node-based smoothed strains. As a very novel and important property, the approach can obtain the upper bound solution in energy norm for hydraulic structures. A 2D gravity dam problem and a 3D arch dam problem are solved, respectively, using the NS-PIM and the simulation results of NS-PIM are found to be the upper bounds. Together with standard fully compatible FEM results as a lower bound, we have successfully determined the solution bounds to certify the accuracy of numerical solutions. This confirms that the NS-PIM is very useful for producing certified solutions for the analysis of huge hydraulic structures. Copyright © 2009 John Wiley & Sons, Ltd. [source]

### Some numerical issues using element-free Galerkin mesh-less method for coupled hydro-mechanical problems

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 7 2009
Abstract A new formulation of the element-free Galerkin (EFG) method is developed for solving coupled hydro-mechanical problems. The numerical approach is based on solving the two governing partial differential equations of equilibrium and continuity of pore water simultaneously. Spatial variables in the weak form, i.e. displacement increment and pore water pressure increment, are discretized using the same EFG shape functions. An incremental constrained Galerkin weak form is used to create the discrete system equations and a fully implicit scheme is used for discretization in the time domain. Implementation of essential boundary conditions is based on a penalty method. Numerical stability of the developed formulation is examined in order to achieve appropriate accuracy of the EFG solution for coupled hydro-mechanical problems. Examples are studied and compared with closed-form or finite element method solutions to demonstrate the validity of the developed model and its capabilities. The results indicate that the EFG method is capable of handling coupled problems in saturated porous media and can predict well both the soil deformation and variation of pore water pressure over time. Some guidelines are proposed to guarantee the accuracy of the EFG solution for coupled hydro-mechanical problems. Copyright © 2008 John Wiley & Sons, Ltd. [source]

### A discrete model for the dynamic propagation of shear bands in a fluid-saturated medium

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 2 2007
Julien Réthoré
Abstract The first part of this manuscript discusses a finite element method that captures arbitrary discontinuities in a two-phase medium by exploiting the partition-of-unity property of finite element shape functions. The fluid flow away from the discontinuity is modelled in a standard fashion using Darcy's relation, and at the discontinuity a discrete analogy of Darcy's relation is used. Subsequently, dynamic shear banding is studied numerically for a biaxial, plane-strain specimen. A Tresca-like as well as a Coulomb criterion is used as nucleation criterion. Decohesion is controlled by a mode-II fracture energy, while for the Coulomb criterion, frictional forces are transmitted across the interface in addition to the cohesive shear tractions. The effect of the different interface relations on the onset of cavitation is studied. Finally, a limited quantitative study is made on the importance of including a so-called dynamic seepage term in Darcy's relation when considering dynamic shear banding. Copyright © 2006 John Wiley & Sons, Ltd. [source]

### An improved meshless collocation method for elastostatic and elastodynamic problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 8 2008
P. H. Wen
Abstract Meshless methods for solving differential equations have become a promising alternative to the finite element and boundary element methods. In this paper, an improved meshless collocation method is presented for use with either moving least square (MLS) or compactly supported radial basis functions (RBFs). A new technique referred to as an indirect derivative method is developed and compared with the direct derivative technique used for evaluation of second-order derivatives and higher-order derivatives of the MLS and RBF shape functions at the field point. As the derivatives are obtained from a local approximation (MLS or compact support RBFs), the new method is computationally economical and efficient. Neither the connectivity of mesh in the domain/boundary nor integrations with fundamental/particular solutions is required in this approach. The accuracy of the two techniques to determine the second-order derivative of shape function is assessed. The applications of meshless method to two-dimensional elastostatic and elastodynamic problems have been presented and comparisons have been made with benchmark analytical solutions. Copyright © 2007 John Wiley & Sons, Ltd. [source]

### Coupling of mesh-free methods with finite elements: basic concepts and test results

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2006
T. Rabczuk
Abstract This paper reviews several novel and older methods for coupling mesh-free particle methods, particularly the element-free Galerkin (EFG) method and the smooth particle hydrodynamics (SPH), with finite elements (FEs). We study master,slave couplings where particles are fixed across the FE boundary, coupling via interface shape functions such that consistency conditions are satisfied, bridging domain coupling, compatibility coupling with Lagrange multipliers and hybrid coupling methods where forces from the particles are applied via their shape functions on the FE nodes and vice versa. The hybrid coupling methods are well suited for large deformations and adaptivity and the coupling procedure is independent of the particle distance and nodal arrangement. We will study the methods for several static and dynamic applications, compare the results to analytical and experimental data and show advantages and drawbacks of the methods. Copyright © 2006 John Wiley & Sons, Ltd. [source]

### Study on the degeneration of quadrilateral element to triangular element

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 9 2004
L.-X. Li
Abstract In this paper, the problems involved in the process of degeneration of quadrilateral element into triangular element are thoroughly analysed. The contents include the formulation of the geometry mapping induced by collapsing one side of the quadrilateral element and the construction of the shape functions. The study focuses first on a 4-node bilinear quadrilateral (Q4) element to 3-node constant strain triangular (CST) element, and then on a 8-node serendipity (Q8) element to 6-node triangular element (T6). In the analysis, the quadrilateral element and degenerate triangular element are assumed to be enclosed by straight edges. The theoretical results show that there is another better approach to realize the degeneration, and that even for conventional approach of degeneration we can give more reasonable explanation to the unclear problems like the CST property in degenerate CST element and the necessity of the additional terms in degenerate T6 element. Copyright © 2004 John Wiley & Sons, Ltd. [source]

### A note on enrichment functions for modelling crack nucleation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2003
J. Bellec
Abstract For particular discretizations and crack configurations, the enhanced approximations of the eXtended finite-element method (X-FEM) cannot accurately represent the discontinuities in the near-tip displacement fields. In this note, we focus on the particular case where the extent of the crack approaches the support size of the nodal shape functions. Under these circumstances, the asymptotic ,branch' functions for each tip may extend beyond the length of the crack, resulting in a non-conforming approximation. We explain the limitations of the standard approximation for arbitrary discontinuities, and propose a set of adjustments to remedy the deficiencies. We also provide numerical results that demonstrate the advantages of the modified approximation. Copyright © 2003 John Wiley & Sons, Ltd. [source]

### Coupling of mapped wave infinite elements and plane wave basis finite elements for the Helmholtz equation in exterior domains

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2003
Rie Sugimoto
Abstract The theory for coupling of mapped wave infinite elements and special wave finite elements for the solution of the Helmholtz equation in unbounded domains is presented. Mapped wave infinite elements can be applied to boundaries of arbitrary shape for exterior wave problems without truncation of the domain. Special wave finite elements allow an element to contain many wavelengths rather than having many finite elements per wavelength like conventional finite elements. Both types of elements include trigonometric functions to describe wave behaviour in their shape functions. However the wave directions between nodes on the finite element/infinite element interface can be incompatible. This is because the directions are normally globally constant within a special finite element but are usually radial from the ,pole' within a mapped wave infinite element. Therefore forcing the waves associated with nodes on the interface to be strictly radial is necessary to eliminate this internode incompatibility. The coupling of these elements was tested for a Hankel source problem and plane wave scattering by a cylinder and good accuracy was achieved. This paper deals with unconjugated infinite elements and is restricted to two-dimensional problems. Copyright © 2003 John Wiley & Sons, Ltd. [source]

### A stabilized pseudo-shell approach for surface parametrization in CFD design problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 4 2002
O. Soto
Abstract A surface representation for computational fluid dynamics (CFD) shape design problems is presented. The surface representation is based on the solution of a simplified pseudo-shell problem on the surface to be optimized. A stabilized finite element formulation is used to perform this step. The methodology has the advantage of being completely independent of the CAD representation. Moreover, the user does not have to predefine any set of shape functions to parameterize the surface. The scheme uses a reasonable discretization of the surface to automatically build the shape deformation modes, by using the pseudo-shell approach and the design parameter positions. Almost every point of the surface grid can be chosen as design parameter, which leads to a very rich design space. Most of the design variables are chosen in an automatic way, which makes the scheme easy to use. Furthermore, the surface grid is not distorted through the design cycles which avoids remeshing procedures. An example is presented to demonstrate the proposed methodology. Copyright © 2002 John Wiley & Sons, Ltd. [source]

### A discontinuous enrichment method for the efficient solution of plate vibration problems in the medium-frequency regime

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2010
Paolo Massimi
Abstract A discontinuous enrichment method (DEM) is presented for the efficient discretization of plate vibration problems in the medium-frequency regime. This method enriches the polynomial shape functions of the classical finite element discretization with free-space solutions of the biharmonic operator governing the elastic vibrations of an infinite Kirchhoff plate. These free-space solutions, which represent flexural waves and decaying modes, are discontinuous across the element interfaces. For this reason, two different and carefully constructed Lagrange multiplier approximations are introduced along the element edges to enforce a weak continuity of the transversal displacement and its normal derivative, and discrete Lagrange multipliers are introduced at the element corners to enforce there a weak continuity of the transversal displacement. The proposed DEM is illustrated with the solution of sample plate vibration problems with different types of harmonic loading in the medium-frequency regime, away from and close to resonance. In all cases, its performance is found to be significantly superior to that of the classical higher-order finite element method. Copyright © 2010 John Wiley & Sons, Ltd. [source]

### A dual mortar approach for 3D finite deformation contact with consistent linearization

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2010
Alexander Popp
Abstract In this paper, an approach for three-dimensional frictionless contact based on a dual mortar formulation and using a primal,dual active set strategy for direct constraint enforcement is presented. We focus on linear shape functions, but briefly address higher order interpolation as well. The study builds on previous work by the authors for two-dimensional problems. First and foremost, the ideas of a consistently linearized dual mortar scheme and of an interpretation of the active set search as a semi-smooth Newton method are extended to the 3D case. This allows for solving all types of nonlinearities (i.e. geometrical, material and contact) within one single Newton scheme. Owing to the dual Lagrange multiplier approach employed, this advantage is not accompanied by an undesirable increase in system size as the Lagrange multipliers can be condensed from the global system of equations. Moreover, it is pointed out that the presented method does not make use of any regularization of contact constraints. Numerical examples illustrate the efficiency of our method and the high quality of results in 3D finite deformation contact analysis. Copyright © 2010 John Wiley & Sons, Ltd. [source]

### A novel singular node-based smoothed finite element method (NS-FEM) for upper bound solutions of fracture problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2010
G. R. Liu
Abstract It is well known that the lower bound to exact solutions in linear fracture problems can be easily obtained by the displacement compatible finite element method (FEM) together with the singular crack tip elements. It is, however, much more difficult to obtain the upper bound solutions for these problems. This paper aims to formulate a novel singular node-based smoothed finite element method (NS-FEM) to obtain the upper bound solutions for fracture problems. In the present singular NS-FEM, the calculation of the system stiffness matrix is performed using the strain smoothing technique over the smoothing domains (SDs) associated with nodes, which leads to the line integrations using only the shape function values along the boundaries of the SDs. A five-node singular crack tip element is used within the framework of NS-FEM to construct singular shape functions via direct point interpolation with proper order of fractional basis. The mix-mode stress intensity factors are evaluated using the domain forms of the interaction integrals. The upper bound solutions of the present singular NS-FEM are demonstrated via benchmark examples for a wide range of material combinations and boundary conditions. Copyright © 2010 John Wiley & Sons, Ltd. [source]